Now briefly describe the reason of using the standard stress-tensor (REF ) (or eq. (REF ) in the main text) in the case of non-trivial acceleration of \(n_a\)  defined by \(a^c=n^b \nabla _b n^c\) . Here we follow [1]} and slightly correct the result mentioned there. The relevant part of the gravitational action variation that includes the \(n_a\)  field acceleration is (cf. eq. (A1) of [2]}):
\(\delta S_{\text{out}}\Big |_{a^c \ne 0}=\int d^3x\,\sqrt{-h}\left[\left(a_a n^a h^{bc}+n^c a^b+n^b a^c\right)\delta h_{bc}- h^{bc} a_c n^a \delta h_{ab}-n^a a^b \delta h_{ab}\right].\) 
